The paper deals with iterative methods for the solution of very large severely ill-conditioned linear systems of equations obtained by the discretization of linear ill-posed problems. Problems are said to be ill-posed according to Hadamard’s definition. Examples are treated. In the equation \(Ax=b\) the right-hand side vector \(b\) represents the given data and is assumed to be contaminated by errors. Known methods employ filtering to reduce the influence of errors and to compute approximate solutions. Often, regularization parameters are used in filtering methods.
In this paper one discusses how the filtering affects the computed approximate solutions. Proposals for the selection of the above mentioned parameters are given. The following methods are considered: the Krylov subspace iterative method, the standard conjugate gradient algorithm, Tikhonov regularization, explicit approximation of the filter function. New iterative methods based on expanded explicitly chosen filter functions in terms of Chebyshev polynomials are presented (Landweber iteration, truncated conjugate gradient iteration). Some remarks on the computation of the regularization parameters are given. Some very nice examples arising the restoration of grey-scale images that have been degraded by blur and noise are presented.
For the entire collection see [Zbl 0934.00020].